8th Grade The Pythagorean Theorem Worksheet Solutions (PDF)
This pythagorean theorem answers for 8th Grade covers the pythagorean theorem (Common Core 8.G.B.6, 8.G.B.7), with a free PDF to print.
This answer key gives a full worked solution for each of the 20 problems on the the Pythagorean theorem worksheet (Common Core 8.G.B.6 and 8.G.B.7). Compare each step with your own work, not only the final answer, and note any step where your method differed.
Grade 8 · Common Core State Standards: 8.G.B.6, 8.G.B.7
Pythagorean theorem answers: exercises 1 to 10
Exercise 1 – Find the hypotenuse
\(6^{2} + 8^{2} = 36 + 64 = 100\).
\(c = \sqrt{100} = 10\).
10 inches
Exercise 2 – Another triple
\(25 + 144 = 169\), so \(c = 13\).
13
Exercise 3 – Find a leg
\(8^{2} + b^{2} = 17^{2}\), so \(b^{2} = 289 – 64 = 225\).
15
Exercise 4 – Not a whole number
\(49 + 81 = 130\).
\(\sqrt{130} \approx 11.4\), because \(11.4^{2} = 129.96\).
about 11.4
Exercise 5 – Is it a right triangle?
\(9^{2} + 12^{2} = 81 + 144 = 225 = 15^{2}\).
Yes
Exercise 6 – Not a right triangle
\(7^{2} + 8^{2} = 113\), but \(12^{2} = 144\).
No
Exercise 7 – Ladder
\(5^{2} + h^{2} = 13^{2}\), so \(h^{2} = 144\).
12 feet
Exercise 8 – Rectangle diagonal
\(12^{2} + 9^{2} = 144 + 81 = 225\).
15 feet
Exercise 9 – TV screen
\(40^{2} + 24^{2} = 1600 + 576 = 2176\).
\(\sqrt{2176} \approx 46.6\).
about 46.6 inches
Exercise 10 – Isosceles right triangle
\(10^{2} + 10^{2} = 200\), so \(c = \sqrt{200} \approx 14.1\).
about 14.1
Exercises 11 and up
Exercise 11 – Perimeter
Hypotenuse: \(\sqrt{81 + 144} = 15\).
Perimeter: \(9 + 12 + 15 = 36\).
36
Exercise 12 – Test the converse
\(20^{2} + 21^{2} = 400 + 441 = 841\) and \(29^{2} = 841\).
Yes, it is a right triangle
Exercise 13 – 3D box
Floor diagonal: \(\sqrt{12^{2} + 4^{2}} = \sqrt{160}\).
Space diagonal: \(\sqrt{160 + 9} = \sqrt{169} = 13\).
13
Exercise 14 – Smaller box
\(2^{2} + 3^{2} + 6^{2} = 4 + 9 + 36 = 49\).
7
Exercise 15 – Baseball diamond
The diagonal splits the square into right triangles: \(90^{2} + 90^{2} = 16{,}200\).
\(\sqrt{16200} \approx 127.3\).
about 127.3 feet
Exercise 16 – Ramp
\(12^{2} + 5^{2} = 169\).
13 feet
Exercise 17 – Kite string
\(30^{2} + 40^{2} = 2500\).
50 feet
Exercise 18 – Shortcut
Direct path: \(\sqrt{25 + 144} = 13\) miles.
Walked: \(5 + 12 = 17\) miles. Saved: \(17 – 13 = 4\).
4 miles
Exercise 19 – Phone screen
\(2.5^{2} + 5.5^{2} = 6.25 + 30.25 = 36.5\).
\(\sqrt{36.5} \approx 6.0\).
about 6.0 inches
Exercise 20 – Explain
Take sides 3, 4 and 6: \(3^{2} + 4^{2} = 25\), but \(6^{2} = 36\). The triangle has no right angle.
The theorem works only for right triangles
Key facts behind this pythagorean theorem answers
- \(a^{2} + b^{2} = c^{2}\), where \(c\) is the hypotenuse.
- To find a leg: \(a^{2} = c^{2} – b^{2}\).
- Converse: if \(a^{2} + b^{2} = c^{2}\), the triangle has a right angle.
- Common triples: 3-4-5, 5-12-13, 8-15-17 and their multiples.
- The theorem works in 3D: the space diagonal satisfies \(d^{2} = a^{2} + b^{2} + c^{2}\).
How to use this pythagorean theorem answers
Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.
All Grade 8 chapters: Grade 8 math.
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